Atiyah Classes in the Context of Generalized Complex Geometry

TL;DR

This paper demonstrates the equivalence of multiple definitions of Atiyah classes for generalized holomorphic vector bundles within generalized complex geometry, unifying different approaches such as Čech cohomology, jet sequences, and Lie algebroids.

Contribution

It establishes the equivalence among three different approaches to defining Atiyah classes in generalized complex geometry, clarifying their relationships.

Findings

Proves the equivalence of Čech cohomology, jet sequence, and Lie algebroid definitions.

Unifies different methods of defining Atiyah classes in generalized complex geometry.

Provides a comprehensive understanding of Atiyah classes in the generalized setting.

Abstract

In analogy to the classical holomorphic setting, Lang, Jia and Liu introduced the notion of the Atiyah class for a generalized holomorphic vector bundle using three different approaches: leveraging $\rm{\check{C}}$ech cohomology, employing the first jet short exact sequence, and adopting the perspective of Lie algebroid pairs. The purpose of this note is to establish the equivalence among these diverse definitions of the Atiyah class.